Study on the Two Right Triangles

Abstract

Suppose that the two right triangles ABC and DEF satisfy the following: ∠A= 90°, AB=AC ; ∠D=90°, ∠F=60° and BC=DE. We consider geometric figures in the situation such as a vertex of one triangle is lapped over some vertex of the other, and one triangle is fixed and the other turns around the lapped vertex. Except for congruence, we pay attention to such geometric figures that satisfy the conditions (1) and (2): (1) The sides of those triangles are not overlapped. (2) A vertex of the fixed triangle lies on a side of the moving triangle, or a vertex of the moving triangle lies on a side of the fixed triangle. In Section 1, we consider one of such geometric figures. In Section 2, we consider the case of A=F and give the explicit expressions of the areas of the overlapped geometric figures. By examining all the cases in Section 3, we can find the twelve geometric figures and we list those figures in Subsection 3.7. This article becomes useful for a motivation expecting and prompting student's mathematical activities, for an example in teaching of ways of seeing and thinking by using mathematics, and for a reference in designing topic-based learnings.